Basic category theory by van Oosten J.
By van Oosten J.
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Exercise 69 Suppose f : X → Y is an arrow in a regular category. For m a subobject M of X, represented by a mono E → X, write ∃f (M ) for the subobject Im(f m) of Y . a) Show that ∃f (M ) is well-defined, that is: depends only on M , not on the representative m. b) Show that if M ∈ Sub(X) and N ∈ Sub(Y ), then ∃f (M ) ≤ N if and only if M ≤ f ∗ (N ). 2 The logic of regular categories The fragment of first order logic we are going to interpret in regular categories is the so-called regular fragment.
One of the points of categorical logic is now, that categorical statements about objects and arrows in C can be reformulated as statements about the truth of certain sequents in L(C). You should read the relevant sequents as expressing that we can “do as if the category were Set”. Examples a) C is a terminal object of C if and only if the sequents ∃x(x = x) are valid, where x, y variables of sort C; b) the arrow f : A → B is mono in C if and only if the sequent f (x) = f (y) x,y x = y is true; c) The square f A x,y x = y and /B g h C /D d is a pullback square in C if and only if the sequents h(xB ) = d(y C ) x,y ∃z A (f (z) = x ∧ g(z) = y) and f (z A ) = f (z A ) ∧ g(z A ) = g(z A ) z,z z=z are true; d) the fact that f : A → B is epi can not similarly be expressed!
2. In chapter 3 we’ve seen that f is epi iff is a pushout; since left adjoints preserve identities and pushouts, they preserve epis; therefore the forgetful functor Mon → Set does not have a right adjoint; b) The functor (−) × X : Set → Set does not preserve the terminal object unless X is itself terminal in Set; therefore, it does not have a left adjoint for non-terminal X. c) The forgetful functor Pos → Set has a left adjoint, but it cannot have a right adjoint: it preserves all coproducts, including the initial object, but not all coequalizers.